Loop dynamics of a fully discrete short pulse equation

TL;DR

This paper introduces a fully discrete short pulse equation as an integrable system, derives semi-discrete versions, and computes explicit multi-soliton and breather solutions, analyzing their dynamics and interactions.

Contribution

It presents a novel fully discrete short pulse equation, derives semi-discrete limits, and explicitly constructs multi-soliton and breather solutions using Darboux transformations.

Findings

Explicit multi-soliton solutions obtained

Breather solutions explicitly derived

Dynamics of loop solitons and interactions analyzed

Abstract

In this article, a fully discrete short pulse (SP) equation is presented as an integrability condition of a linear system of difference equations (also known as discrete Lax pair). Additionally, two semi-discrete versions of the SP equation have also been obtained from fully discrete SP equation under continuum limits. Darboux transformation is employed to compute multi-soliton solutions of fully discrete and semi-discrete SP equations. Explicit expressions of first and second nontrivial soliton solutions are computed. We also derived explicit expression of breather solution for fully discrete SP equation. The dynamics of single loop soliton and interaction mechanism of loop-loop and loop-antiloop solutions has been explored and illustrated.